ANALYSIS part of the Exercise Set 4
1&2. Reading the Boston data from the MASS package and exploring it
library(MASS)
data("Boston")
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...dim(Boston)## [1] 506 14This dataset contains information collected by the U.S Census Service concerning housing in the area of Boston Mass. It was obtained from the StatLib archive (http://lib.stat.cmu.edu/datasets/boston), and has been used extensively throughout the literature to benchmark algorithms. However, these comparisons were primarily done outside of Delve and are thus somewhat suspect. The dataset is small in size with only 506 cases.The data was originally published by Harrison, D. and Rubinfeld, D.L. `Hedonic prices and the demand for clean air’, J. Environ. Economics & Management, vol.5, 81-102, 1978.
3. Variables of Boston data.
There are 14 attributes in each case of the dataset. They are: CRIM - per capita crime rate by town; ZN - proportion of residential land zoned for lots over 25,000 sq.ft.; INDUS - proportion of non-retail business acres per town; CHAS - Charles River dummy variable (1 if tract bounds river; 0 otherwise); NOX - nitric oxides concentration (parts per 10 million); RM - average number of rooms per dwelling; AGE - proportion of owner-occupied units built prior to 1940; DIS - weighted distances to five Boston employment centres; RAD - index of accessibility to radial highways; TAX - full-value property-tax rate per $10,000; PTRATIO - pupil-teacher ratio by town; B - 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town; LSTAT - % lower status of the population; MEDV - Median value of owner-occupied homes in $1000’s.
Let’ see the matrix of Boston variables.
pairs(Boston)
summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00Let’s see correlations between variables in the dataset.
library(dplyr)##
## Attaching package: 'dplyr'## The following object is masked from 'package:MASS':
##
## select## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(corrplot)## corrplot 0.84 loadedcor_matrix<-cor(Boston) %>% round(digits = 2)
corrplot(cor_matrix, method="circle", type="upper", cl.pos="b", tl.pos="d", tl.cex = 0.6)
Interpretation: we can see that some variables correlate greatly with ech other, for example, INDUS, NOX and AGE (positive correlation). INDUS and NOX correlate negatively. COrrelation plot is very informative about the relationships of the variables.
4.Standardization the dataset using scale() and summaries of the scaled data.
boston_scaled <- scale(Boston)
summary(boston_scaled)## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865class(boston_scaled)## [1] "matrix"boston_scaled <- as.data.frame(boston_scaled)
class(boston_scaled)## [1] "data.frame"As we can see now, mean of each variable is set to zero.We also converted boston_scale object from matrix to the dateframe using R functions and verified the class of the object.
Now we will create a categorical variable of the crime rate from the scaled crime rate using the quantiles as the break points in the categorical variable. We need to drop the old crime rate variable from the dataset, too.
Let’s have a look at the variable “crime”.
summary(boston_scaled$crim)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.419367 -0.410563 -0.390280 0.000000 0.007389 9.924110Now we make a quantile vector of it and check it.
bins <- quantile(boston_scaled$crim)
bins## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610Now we make a categorical variable out of variable “crime” with 4 labels.Then we check frequences by table().
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, labels = c("low", "med_low", "med_high", "high"))
table(crime)## crime
## low med_low med_high high
## 127 126 126 127Next step is to remove original crim from the dataset and add the new categorical value to scaled data. We will also check it afterwards.
boston_scaled <- dplyr::select(boston_scaled, -crim)
boston_scaled <- data.frame(boston_scaled, crime)
table(boston_scaled$crime)##
## low med_low med_high high
## 127 126 126 127Now we will create a train dataset(80% of the dataset) and a test dataset from Boston.
n <- nrow(boston_scaled)
ind <- sample(n, size = n * 0.8)
train <- boston_scaled[ind,]
test <- boston_scaled[-ind,]5.Linear discriminant analysis
Let’s run linear discriminant analysis.
lda.fit <- lda(crime ~ ., data = train)
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2549505 0.2450495 0.2524752 0.2475248
##
## Group means:
## zn indus chas nox rm
## low 1.0640380 -0.9036420 -0.15765625 -0.8819051 0.4110106
## med_low -0.1712920 -0.2584181 -0.03371693 -0.5239687 -0.1556059
## med_high -0.3939194 0.2175024 0.22945822 0.4046783 0.1561290
## high -0.4872402 1.0171519 -0.03610305 1.1087118 -0.4566418
## age dis rad tax ptratio
## low -0.9069034 0.8733585 -0.6975420 -0.7332905 -0.468721678
## med_low -0.2739818 0.2647897 -0.5468458 -0.4575606 0.004675854
## med_high 0.4274056 -0.3920281 -0.3817409 -0.2771944 -0.289661896
## high 0.8238024 -0.8676900 1.6377820 1.5138081 0.780373633
## black lstat medv
## low 0.37954843 -0.758971107 0.50070737
## med_low 0.31733792 -0.113305822 -0.01619727
## med_high 0.04488567 -0.003202208 0.20301974
## high -0.77278971 0.901999899 -0.63844078
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.0880161075 0.88509974 -0.88291726
## indus 0.0115844326 -0.24837392 0.13785851
## chas -0.0930756421 -0.07843240 0.09532990
## nox 0.4782700500 -0.62191412 -1.45600112
## rm -0.0928351313 -0.11964937 -0.19966134
## age 0.3040160398 -0.33863490 -0.06823650
## dis 0.0137669211 -0.50007345 0.18047576
## rad 2.9795691185 1.05506386 -0.06350704
## tax 0.0001341613 -0.20975161 0.69074677
## ptratio 0.1311160608 0.01648687 -0.17549334
## black -0.1285281124 0.01862796 0.11207535
## lstat 0.2386136476 -0.28649024 0.36368632
## medv 0.2470427036 -0.41113274 -0.16840048
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9403 0.0439 0.0158Now we will plot the results of lda.
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "orange", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
classes <- as.numeric(train$crime)
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 1)
6. classes prediction with the LDA model on the test data. Cross-tabulation of the results.
correct_classes <- test$crime
test <- dplyr::select(test, -crime)
lda.pred <- predict(lda.fit, newdata = test)
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## low 12 11 1 0
## med_low 9 17 1 0
## med_high 1 13 10 0
## high 0 0 0 277. Reload of Boston and k-means algorithm.
library(MASS)
data("Boston")
dist_eu <- dist(scale(Boston))
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970dist_man <- dist(scale(Boston), method = "manhattan")
summary(dist_man)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2662 8.4832 12.6090 13.5488 17.7568 48.8618Now we do k-means clustering and plot it!
km <-kmeans(Boston, centers = 3)
pairs(Boston, col = km$cluster)
Next step is to investigate k number.
library(ggplot2)
k_max <- 10
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})
qplot(x = 1:k_max, y = twcss, geom = 'line')
km <-kmeans(Boston, centers = 2)
pairs(Boston, col = km$cluster)
Interpretation: The optimal number of clusters is when the value of total WCSS changes radically. In this case, two clusters would seem optimal.SO we reran k-means with two clusters.
Super-Bonus.
model_predictors <- dplyr::select(train, -crime)
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:MASS':
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## select## The following object is masked from 'package:stats':
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## filter## The following object is masked from 'package:graphics':
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## layoutplot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers')Nice plot!